Effects of a Vanishing Noise on Elementary Cellular Automata Phase-Space Structure

TL;DR

This work obtains the complete phase space of all minimal ECA, and investigates how a vanishing noise alters this phase space, connecting attractors and modifying the asymptotic probability distribution.

Abstract

We investigate elementary cellular automata (ECA) from the point of view of (discrete) dynamical systems. By studying small lattice sizes, we obtain the complete phase space of all minimal ECA, and, starting from a maximal entropy distribution (all configurations equiprobable), we show how the dynamics affects this distribution. We then investigate how a vanishing noise alters this phase space, connecting attractors and modifying the asymptotic probability distribution. What is interesting is that this modification not always goes in the sense of decreasing the entropy.

Figures (6)

• Figure 1: Time evolution of some rules starting from a random initial configuration ($s_i(t)$) and the evolution of an initial single defect ($S_i(t)$). Time runs from top to down. Color code: black: $s_i(t)=S_i(t)=0$; blue: $s_i(t)=1$, $S_i(t)=0$; green: $s_i(t)=S_i(t)=1$; red: $s_i(t)=0$, $S_i(t)=1$.
• Figure 2: Transients and attractors for a basin of Rule 30, $L=5$. For this length, Rule 30 has another attractor, the fixed point $00000$ whose basin also includes the configuration $11111$.
• Figure 3: Attractors of rules shown in Fig. \ref{['fig:evolv']}. Depending on the rule the dynamics shows different behaviors; for example, fixed point attractors (e.g., bottom left of rule 1), cycles with/without a basin of attraction (e.g., rule 30 and 150).
• Figure 4: The ideal phase space of the Hopfield model with $L=4$ and $K=6$ stored patterns. In this case all stored patterns and their inverses are attractors of the dynamics (black arrows). The effect of noise (red arrows) is that of connecting all attractors.
• Figure 5: Instances of the phase space of the Hopfield model with dimension $L=4$ and number of stored patterns $K=6$. In black the transitions given by the dynamics, Eq. \ref{['eq:sigma']}, in red the transitions induced by the vanishing noise. Nodes are indicated by configuration (expressing the number in base-two representation) and energy in parenthesis. The attractors are marked by green dots. (a) Two resulting attractors, both stable with respect to the noise. The entropy is $S=S^*=0.5$. (b) Four attractors, all with the same basin size. The noise connects attractors but does not affect the entropy ($S=S^*=0.25$). (c) Four attractors with different basin size. Two of them are unstable with respect to noise, which reduces the entropy from $S=0.386$ to $S^*=0.250$. (d) Four attractors, with different basin size. The entropy connects the attractors flattening the probability distributions. The entropy increases from $S=0.489$ to $S^*=493$.